{"id":2334,"date":"2022-02-25T18:04:59","date_gmt":"2022-02-25T09:04:59","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2334"},"modified":"2025-06-11T10:01:43","modified_gmt":"2025-06-11T01:01:43","slug":"2%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%ae%e8%a8%88%e7%ae%97%ef%bc%9a%e7%b4%af%e6%ac%a1%e7%a9%8d%e5%88%86%e3%81%ae%e3%81%be%e3%81%a8%e3%82%81","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/2%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%ae%e8%a8%88%e7%ae%97%ef%bc%9a%e7%b4%af%e6%ac%a1%e7%a9%8d%e5%88%86%e3%81%ae%e3%81%be%e3%81%a8%e3%82%81\/","title":{"rendered":"2\u91cd\u7a4d\u5206\u306e\u8a08\u7b97\uff1a\u7d2f\u6b21\u7a4d\u5206\u306e\u307e\u3068\u3081"},"content":{"rendered":"

\\(\\displaystyle\\iint_D f(x,y)\\,dx\\, dy\\) \u306e\u9818\u57df \\(D\\) \u304c\u4f55\u3068\u4f55\u3067\u56f2\u307e\u308c\u3066\u3044\u308b\u304b\u3092\u660e\u3089\u304b\u306b\u3057\u3066\uff0c$x, y$ \u3069\u3061\u3089\u306e\u7a4d\u5206\u5909\u6570\u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u304b\u3092\u6c7a\u3081\u3066\u7a4d\u5206\u3059\u308b\u3002$x$ \u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u969b\u306b\u306f\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $y$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $x$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002\u9006\u306b\uff0c$y$ \u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u969b\u306b\u306f\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $x$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $y$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002\u3053\u308c\u304c2\u91cd\u7a4d\u5206\u3092\u7d2f\u6b21\u7a4d\u5206\u306e\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u304d\u306e\u3084\u308a\u304b\u305f\u3002
\n<\/p>\n

\n

\u9818\u57df $D$ \u304c $x = a, \\ x = b, \\ y = c, \\ y = d$ \u3067\u56f2\u307e\u308c\u308b\u5834\u5408<\/h3>\n

\\begin{eqnarray}
\n\\iint_D f(x,y) \\,dx\\, dy &=& \\int_a^b \\left\\{\\int_c^d f(x,y)\\,dy\u00a0 \\right\\} \\,dx\u00a0 \\\\
\n&=& \\int_c^d \\left\\{ \\int_a^bf(x,y)\\,dx\u00a0 \\right\\} \\,dy
\n\\end{eqnarray}<\/p>\n

\u7a4d\u5206\u306e\u969b\uff0c\u4f8b\u3048\u3070\u5185\u5074\u306e $\\displaystyle \\int_c^d f(x,y)\\,dy $ \u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u306f\uff0c$y$ \u306e\u307f\u306b\u3064\u3044\u3066\u5148\u306b\u7a4d\u5206\u3067\u3042\u308b\u304b\u3089\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $x$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $y$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002<\/strong><\/span><\/span>\u540c\u69d8\u306b\uff0c\u5185\u5074\u306e $\\displaystyle \\int_a^bf(x,y)\\,dx $ \u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u306f\uff0c$x$ \u306e\u307f\u306b\u3064\u3044\u3066\u5148\u306b\u7a4d\u5206\u3067\u3042\u308b\u304b\u3089\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $y$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $x$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002<\/strong><\/span><\/span>\u3053\u308c\u304c2\u91cd\u7a4d\u5206\u3092\u7d2f\u6b21\u7a4d\u5206\u306e\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u304d\u306e\u3084\u308a\u304b\u305f\u3002<\/p>\n

\u9818\u57df \\(D\\) \u304c \\(x = a, \\ x = b, \\ y = y_1(x), \\ y = y_2(x)\\) \u3067\u56f2\u307e\u308c\u308b\u5834\u5408<\/h3>\n

\"\"<\/p>\n

$$\\iint_D f(x,y) \\,dx\\, dy = \\int_a^b \\left\\{\\int_{y_1(x)}^{y_2(x)} f(x,y)\\,dy \\right\\}\\, dx $$<\/p>\n

\u5148\u306b $\\displaystyle \\int_{y_1(x)}^{y_2(x)} f(x,y)\\,dy$ \u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u306f\uff0c$y$ \u306e\u307f\u306b\u3064\u3044\u3066\u5148\u306b\u7a4d\u5206\u3059\u308b\u306e\u3067\u3042\u308b\u304b\u3089\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $x$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $y$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002<\/strong><\/span><\/span>\u3053\u308c\u304c2\u91cd\u7a4d\u5206\u3092\u7d2f\u6b21\u7a4d\u5206\u306e\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u304d\u306e\u3084\u308a\u304b\u305f\u3002<\/p>\n

\u89e3\u8aac\uff1a<\/h4>\n

\\begin{eqnarray}\\iint_D f(x,y)\\, dx\\, dy &=&
\n\\int_a^b \\left\\{ \\int_{y_1(x)}^{y_2(x)} f(x, y) dy \\right\\}\u00a0 \\,dx \\quad(\\mbox{$\\{\\ \\}$ \u306e\u4e2d\u3092\u5148\u306b\u7a4d\u5206})\\\\
\n&=& \\int_a^b \\int_{y_1(x)}^{y_2(x)} f(x, y)\\, dy \\,dx \\quad(\\mbox{\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u8fd1\u3044 $dy$ \u3092\u5148\u306b\u7a4d\u5206})\\\\
\n&=& \\int_a^b dx \\int_{y_1(x)}^{y_2(x)} dy \\,f(x, y) \\quad(\\mbox{\u88ab\u7a4d\u5206\u95a2\u6570\u306b\u8fd1\u3044 $dy$ \u3092\u5148\u306b\u7a4d\u5206})
\n\\end{eqnarray}<\/p>\n

\u9ad8\u6821\u3067\u306f\uff0c\u30a4\u30f3\u30c6\u30b0\u30e9\u30eb \\(\\displaystyle \\int\\) \u3068 \\(dx\\) \u306e\u9593\u306b\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x)\\) \u3092\u631f\u3093\u3067\u66f8\u3044\u3066\u307e\u3057\u305f\u304c\uff0c2\u91cd\u7a4d\u5206\u306e\u3088\u3046\u306a\u5834\u5408\u306f\uff0c\u7a4d\u5206\u7bc4\u56f2\u304c\uff0c\\(x\\) \u306e\u7bc4\u56f2\u306a\u306e\u304b \\(y\\) \u306e\u7bc4\u56f2\u306a\u306e\u304b\u6df7\u4e71\u3059\u308b\u5834\u5408\u3082\u3042\u308b\u3067\u3057\u3087\u3046\u3002\u305f\u3068\u3048\u3070<\/p>\n

$$\\int_a^b \\int_c^d f(x, y)\\, dx\\, dy$$<\/p>\n

\u3068\u66f8\u3044\u305f\u3068\u304d\uff0c$[a, b]$ \u306f $x$ \u306e\u7a4d\u5206\u7bc4\u56f2\u306a\u306e\u304b $y$ \u306e\u7a4d\u5206\u7bc4\u56f2\u306a\u306e\u304b\uff0c\u4e00\u77ac\u60a9\u307f\u307e\u305b\u3093\u304b\uff1f<\/p>\n

\u305d\u3053\u3067\uff0c\u4e0a\u5f0f\u306e\u4e00\u756a\u6700\u5f8c\u306e\u8868\u8a18\u306e\u3088\u3046\u306b\uff0c\\(\\displaystyle \\int_a^b dx\\) \u3068\u66f8\u3044\u3066 \\(x\\) \u306e\u7a4d\u5206\u7bc4\u56f2\u304c\\([a, b]\\) \u306a\u3093\u3067\u3059\u3088\u3068\u308f\u304b\u308a\u3084\u3059\u304f\u3059\u308b\u3068\u3044\u3046\u306e\u304c\u5927\u5b66\u7684\uff08\u3061\u3087\u3063\u3068\u901a\u3076\u3063\u305f\uff09\u66f8\u304d\u65b9\u3067\u3059\u3002<\/p>\n

\u7a4d\u5206\u3059\u308b\u5909\u6570\u3068\u305d\u306e\u7a4d\u5206\u7bc4\u56f2\u306e\u30da\u30a2\u3092\u6700\u512a\u5148\u306b
\n$$\\int_a^b dx \\int_{y_1(x)}^{y_2(x)} dy\\, f(x, y) $$ \u306e\u3088\u3046\u306b\u66f8\u304f\u306e\u3067\uff0c\u809d\u5fc3\u306e\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x,y)\\) \u304c\u4e00\u756a\u6700\u5f8c\u306b\u73fe\u308c\u3066\uff0c\u898b\u6163\u308c\u306a\u3044\u3046\u3061\u306f\u4f55\u3068\u306a\u304f\u843d\u3061\u7740\u304b\u306a\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\uff0c\u307e\u3042\uff0c\u6163\u308c\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

2\u91cd\u7a4d\u5206\uff08\u4e00\u822c\u306b\u591a\u91cd\u7a4d\u5206\uff09\u306e\u9806\u756a\u306f\uff0c\u88ab\u7a4d\u5206\u95a2\u6570 $f(x, y)$ \u306b\u8fd1\u3044\u5909\u6570\u306e\u307b\u3046<\/strong><\/span>\uff08\u4e0a\u306e\u4f8b\u3067\u306f $\\displaystyle \\int \\, dy$\uff09\u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b<\/strong><\/span>\u3093\u3067\u3059\u3088\u3002<\/p>\n

 <\/p>\n

\u9818\u57df \\(D\\) \u304c \\(y = c, \\ y = d, \\ x = x_1(y), \\ x = x_2(y)\\) \u3067\u56f2\u307e\u308c\u308b\u5834\u5408<\/h3>\n

\"\"<\/p>\n

 <\/p>\n

$$\\iint_D f(x,y)\\, dx\\, dy = \\int_c^d \\left\\{\\int_{x_1(y)}^{x_2(y)} f(x,y)\\,dx \\right\\}\\, dy\u00a0 = \\int_c^d dy \\int_{x_1(y)}^{x_2(y)} dx\\, f(x,y)$$<\/p>\n

\u5148\u306b $\\displaystyle \\int_{x_1(y)}^{x_2(y)} dx\\, f(x,y)$ \u3092\u8a08\u7b97\u3059\u308b\u969b\u306b\u306f\uff0c$x$ \u306e\u307f\u306b\u3064\u3044\u3066\u5148\u306b\u7a4d\u5206\u3059\u308b\u306e\u3067\u3042\u308b\u304b\u3089\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $y$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $x$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002<\/strong><\/span><\/span>\u3053\u308c\u304c2\u91cd\u7a4d\u5206\u3092\u7d2f\u6b21\u7a4d\u5206\u306e\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u304d\u306e\u3084\u308a\u304b\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\\(\\displaystyle\\iint_D f(x,y)\\,dx\\, dy\\) \u306e\u9818\u57df \\(D\\) \u304c\u4f55\u3068\u4f55\u3067\u56f2\u307e\u308c\u3066\u3044\u308b\u304b\u3092\u660e\u3089\u304b\u306b\u3057\u3066\uff0c$x, y$ \u3069\u3061\u3089\u306e\u7a4d\u5206\u5909\u6570\u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u304b\u3092\u6c7a\u3081\u3066\u7a4d\u5206\u3059\u308b\u3002$x$ \u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u969b\u306b\u306f\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $y$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $x$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002\u9006\u306b\uff0c$y$ \u304b\u3089\u5148\u306b\u7a4d\u5206\u3059\u308b\u969b\u306b\u306f\uff0c\u5f8c\u304b\u3089\u7a4d\u5206\u3059\u308b\u5909\u6570 $x$ \u306f\u56fa\u5b9a\u3057\u3066\uff0c\u3042\u305f\u304b\u3082\u5b9a\u6570\u306e\u3088\u3046\u306b\u6271\u3063\u3066 $y$ \u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3002\u3053\u308c\u304c2\u91cd\u7a4d\u5206\u3092\u7d2f\u6b21\u7a4d\u5206\u306e\u65b9\u6cd5\u3067\u8a08\u7b97\u3059\u308b\u3068\u304d\u306e\u3084\u308a\u304b\u305f\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2228,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2334","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2334","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2334"}],"version-history":[{"count":12,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2334\/revisions"}],"predecessor-version":[{"id":10485,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2334\/revisions\/10485"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2228"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}